where the superscript \(r\)  in \(z_{i}^{(r)}\)  denotes the
number of consensus rounds and \(\epsilon _{i}(t)\)  is the consensus
error at worker \(i\) . The optimization parameter \(w_{i}(t)\)  is updated
according to (REF ). Let \(z(t)\)  denote the dual
variable with perfect consensus. According to Lemma 1 from our previous
work [1]}, let \(\delta \ge 0\)  and \(\lambda _{2}(Q)\)  be
the second largest eigenvalue of \(Q\) , then
\(\Vert z_{i}^{(r)}(t+1)-z(t+1)\Vert \le \delta \) 
